The cyclic coloring complex of a complete k-uniform hypergraph

TL;DR

This paper investigates the homology of cyclic coloring complexes of complete, star, and diagonal hypergraphs, revealing that their homology group dimensions are expressed by binomial coefficients, with implications for cyclic homology of related algebraic structures.

Contribution

It provides explicit formulas for the homology group dimensions of cyclic coloring complexes for various hypergraph types, extending known results and connecting to algebraic structures.

Findings

Homology dimensions for complete hypergraphs are binomial coefficients.

Homology dimensions for star and diagonal hypergraphs are binomial coefficients.

Results relate to the cyclic homology of certain quotient algebras.

Abstract

In this paper, we study the homology of the cyclic coloring complex of three different types of $k$-uniform hypergraphs. For the case of a complete $k$-uniform hypergraph, we show that the dimension of the $(n-k-1)^{st}$ homology group is given by a binomial coefficient. Further, we discuss a complex whose $r$-faces consist of all ordered set partitions $[B_1, ..., B_{r+2}]$ where none of the $B_i$ contain a hyperedge of the complete $k$-uniform hypergraph $H$ and where $1 \in B_1$. It is shown that the dimensions of the homology groups of this complex are given by binomial coefficients. As a consequence, this result gives the dimensions of the multilinear parts of the cyclic homology groups of $\C[x_1,...,x_n]/ \{x_{i_1}...x_{i_k} \mid i_{1}...i_{k}$ is a hyperedge of $H \}$. For the other two types of hypergraphs, star hypergraphs and diagonal hypergraphs, we show that the dimensions of the homology groups of their cyclic coloring complexes are given by binomial coefficients as well.